Categorical smooth compactifications and neighborhoods of infinity

Introduction

This note is devoted to a short overview of some results on derived categories of coherent sheaves concerning smooth categorical compactifications and the formal punctured neighborhoods of infinity.

In Section ⁠1⁠, we discuss the conjecture of Bondal and Orlov about the categorical properties of the resolution of singularities of an algebraic variety with rational singularities (Conjecture ⁠1.1⁠). This conjecture states that the derived pushforward functor on the derived categories of coherent sheaves is a quotient functor (that is, a localization). The conjecture is difficult and still open in general. It turns out that it is possible (Theorem ⁠1.2⁠) to prove a version of such statement for an arbitrary separated scheme of finite type over a field of characteristic zero (the reader may safely assume that we are dealing with quasi-projective schemes). The methods make it possible to prove Conjecture ⁠1.1⁠ for a cone over a projective embedding of a smooth Fano variety (that is, a smooth projective variety with an ample anti-canonical line bundle).

In Section ⁠2⁠, we consider DG categorical smooth compactifications. Here DG stands for “differential-graded”. This is a straightforward generalization of the usual algebro-geometric smooth compactification. The following natural question was formulated by B. Toën (Question ⁠2.3⁠ below): is it true that any smooth DG category “of finite type” admits a smooth categorical compactification? The question was considered to be difficult, but most experts expected that the answer should be “yes”. However, in [⁠4⁠ A. I. Efimov, Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. Math.222, 667–694 (2020) ] we gave a negative answer, obtained by disproving a closely related conjecture of Kontsevich (Conjecture ⁠2.5⁠ below) on the generalized version of the degeneration of the Hodge-to-de-Rham spectral sequence. We also obtained a dual version of these results, in which smooth DG categories are replaced by proper DG categories, and a smooth compactification is replaced by a categorical resolution of singularities.

In Section ⁠3⁠ we outline a certain construction called a “categorical formal punctured neighborhood of infinity”. For a smooth algebraic variety $X$ this is obtained as follows: take some smooth compactification $\overline{X},$ consider the formal completion at the infinity locus $\overline{X}-X,$ and then take the corresponding punctured formal scheme. The resulting object $X_{\widehat{\infty}}$ (considered for example as an adic space) is independent of the compactification, as is the category of perfect complexes on it. In [⁠3⁠ A. I. Efimov, Categorical formal punctured neighborhood of infinity, I. arXiv:1711.00756 (2017) ] we give a purely categorical construction of $\operatorname{Perf}(X_{\widehat{\infty}})$ which generalizes to arbitrary smooth DG algebras and DG categories. A curious special case is the algebra of rational functions on a smooth projective curve. There, our construction gives exactly the ring of adeles.

1 Rational singularities and a conjecture of Bondal and Orlov

Let $X$ be an algebraic variety over a field of characteristic zero. Recall that $X$ has rational singularities if for some (and then any) resolution of singularities $\pi:Y\to X$ we have ${\mathbf{R}}\pi_{*}{\mathcal{O}}_{Y}\cong{\mathcal{O}}_{X}.$ Equivalently, the pullback functor $\mathbb{L}\pi^{*}:D_{\operatorname{perf}}(X)\to D_{\operatorname{perf}}(Y)$ is fully faithful. The following conjecture is still open.

The following result is a version of such statement which holds in a much more general framework.

This theorem in particular confirms a conjecture of Kontsevich on the homotopy finiteness of the DG category $D^{b}_{\mathrm{coh}}(X).$ The proof is based on a certain construction of a categorical resolution of singularities, due to Kuznetsov and Lunts [⁠8⁠ A. Kuznetsov and V. A. Lunts, Categorical resolutions of irrational singularities. Int. Math. Res. Not. IMRN, 4536–4625 (2015) ].

The methods developed to prove Theorem ⁠1.2⁠ actually also work to prove Conjecture ⁠1.1⁠ in a certain class of cases. In particular, the following result holds.

2 Categorical smooth compactifications

Theorem ⁠1.2⁠ deals with a special case of a categorical smooth compactification. We first recall some basic definitions.

In particular, we have the notions of smoothness and properness for DG algebras (a DG algebra can be considered as a DG category with a single object). When $X$ is a separated scheme of finite type over a field $\mathrm{k},$ then $X$ is smooth (resp. proper) if and only if the DG category $\operatorname{Perf}(X)$ is smooth (resp. proper) ([⁠11⁠ D. Orlov, Smooth and proper noncommutative schemes and gluing of DG categories. Adv. Math.302, 59–105 (2016) , Proposition 3.30], [⁠10⁠ V. A. Lunts, Categorical resolution of singularities. J. Algebra323, 2977–3003 (2010) , Proposition 3.13]). Hence, these basic geometric properties of $X$ are reflected by the DG category $\operatorname{Perf}(X).$

We recall the following definition.

The basic geometric example of a categorical smooth compactification is given by the usual one. Namely, let $X$ be a smooth algebraic variety over $\mathrm{k},$ and let $j:X\hookrightarrow\overline{X}$ be an open embedding, where $\overline{X}$ is smooth and proper. Then the restriction functor $j^{*}:\operatorname{Perf}(\overline{X})\to\operatorname{Perf}(X)$ is a categorical smooth compactification.

Theorem ⁠1.2⁠ provides a categorical smooth compactification of the DG categories of the form $D^{b}_{\mathrm{coh}}(X),$ where $X$ is a separated scheme of finite type over a field of characteristic zero.

There is a notion of a homotopically finitely presented (hfp) DG category which should be thought of as a smooth DG category "of finite type" (we refer to [⁠14⁠ B. Toën and M. Vaquié, Moduli of objects in DG-categories. Ann. Sci. École Norm. Sup. (4)40, 387–444 (2007) ] for the precise definition). The following general question was formulated by Bertrand Toën.

The question is difficult, but the general consensus was that the answer should be “yes”. However, in [⁠4⁠ A. I. Efimov, Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. Math.222, 667–694 (2020) ] the author gave a negative answer to this question. Here we explain the rough idea of the results of [⁠4⁠ A. I. Efimov, Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. Math.222, 667–694 (2020) ].

It turns out that Question ⁠2.3⁠ is closely related with the non-commutative (categorical) Hodge-to-de Rham degeneration. Recall that the classical Hodge theory implies (via GAGA) the following algebraic statement: for any smooth algebraic variety $X$ over a field $\mathrm{k}$ of characteristic zero the spectral sequence

degenerates.

The following categorical generalization was conjectured by Kontsevich and Soibelman [⁠7⁠ M. Kontsevich and Y. Soibelman, Notes on A∞-algebras, A∞-categories and non-commutative geometry. In Homological Mirror Symmetry, Lecture Notes in Phys. 757, Springer, Berlin, 153–219 (2009) ], and proved by Kaledin [⁠6⁠ D. Kaledin, Spectral sequences for cyclic homology. In Algebra, Geometry, and Physics in the 21st Century, Progr. Math. 324, Birkhäuser/Springer, Cham, 99–129 (2017) ].

In the special case when $\operatorname{Perf}(A)\simeq\operatorname{Perf}(X)$ for a smooth and proper variety $X,$ Theorem ⁠2.4⁠ gives exactly the usual (commutative) Hodge-to-de Rham degeneration.

The following two conjectures were formulated by Kontsevich for smooth and for proper DG algebras.

Here $\delta^{-}:HH_{\bullet}(A^{\mathrm{op}})\to HC_{\bullet}^{-}(A^{\mathrm{op}})[-1]$ denotes the boundary map, see [⁠2⁠ A. I. Efimov, Generalized non-commutative degeneration conjecture. Proc. Steklov Inst. Math.290, 1–10 (2015) , Section 3].

Here $\delta^{+}:HC_{\bullet}(B^{\mathrm{op}})[1]\to HH_{\bullet}(B^{\mathrm{op}})$ denotes the boundary map, see [⁠9⁠ J.-L. Loday, Cyclic Homology. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 301, Springer, Berlin (1992) , Section 2.2].

Both conjectures ⁠2.5⁠ and ⁠2.6⁠ hold, roughly speaking, for all DG categories coming from (commutative) algebraic geometry.

Conjecture ⁠2.5⁠ is related to Question ⁠2.3⁠ as follows. Suppose that we have a smooth compactification ${\mathcal{C}}\to{\mathcal{A}}$ (hence ${\mathcal{A}}$ is smooth). Then we have the following commutative diagram:

The left vertical map sends $\operatorname{ch}(I_{{\mathcal{C}}})$ to $\operatorname{ch}(I_{{\mathcal{A}}}).$ Hence, applying Kaledin’s Theorem ⁠2.4⁠, we obtain that Conjecture ⁠2.5⁠ holds for ${\mathcal{A}}$.

A dual argument implies that Conjecture ⁠2.6⁠ holds for proper DG categories which can be fully faithfully embedded into a smooth and proper DG category (such an embedding is called a categorical resolution in the terminology of Kuznetsov and Lunts [⁠8⁠ A. Kuznetsov and V. A. Lunts, Categorical resolutions of irrational singularities. Int. Math. Res. Not. IMRN, 4536–4625 (2015) ]).

However, in [⁠4⁠ A. I. Efimov, Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. Math.222, 667–694 (2020) ] we disproved both conjectures.

The DG algebra $B$ from part 2 is quasi-isomorphic to a certain explicit $10$-dimensional $A_{\infty}$-algebra for which the supertrace of $m_{3}$ on the second argument is non-zero.

3 Categorical formal punctured neighborhood of infinity

Another subject related to the notion of a smooth categorical compactification is that of a formal punctured neighborhood of infinity. Suppose that we have a usual smooth compactification $j:X\hookrightarrow\overline{X}$ of a smooth algebraic variety $X$. Then one can take the formal neighborhood $\overline{X}_{\widehat{Z}},$ and then “remove” $Z$. The resulting object $\overline{X}_{\widehat{Z}}-Z$ (the so-called generic fiber, considered as an adic space) does not depend on the choice of the compactification $\overline{X}.$ Let us set $X_{\widehat{\infty}}:=\overline{X}_{\widehat{Z}}-Z.$ The corresponding category of perfect complexes $\operatorname{Perf}(X_{\widehat{\infty}})$ also does not depend on $Z$ and it is therefore an invariant of $X$.

The natural question arises: can we describe the category $\operatorname{Perf}(X_{\widehat{\infty}})$ purely in terms of $\operatorname{Perf}(X)$? This question is partially motivated by mirror symmetry since an analogue of $\operatorname{Perf}(X_{\widehat{\infty}})$ exists in symplectic geometry in the framework of Fukaya categories. It turns out that the purely categorical construction is possible, and it was described by the author in [⁠3⁠ A. I. Efimov, Categorical formal punctured neighborhood of infinity, I. arXiv:1711.00756 (2017) ]. Here we give an outline.

First, we describe a “non-derived” version of the construction. Let $A$ be an associative algebra over a field $\mathrm{k}.$ Then one can describe the algebra $H^{0}(A_{\widehat{\infty}})$ as follows.

Here $\operatorname{End}_{\mathrm{k}}(A)$ is the algebra of $\mathrm{k}$-linear endomorphisms of $A$ (as a vector space) and $A^{*}\otimes A\subset\operatorname{End}_{\mathrm{k}(A)}$ is the two-sided ideal of operators of finite rank. The commutator is the additive one (the Lie algebra bracket) and $R_{a}:A\to A,$$R_{a}(b)=ba,$ is the operator of right multiplication by $a.$

Now let $A$ be a smooth DG algebra. The DG algebra $A_{\widehat{\infty}}$ is defined by the formula

Here $C^{\bullet}(A,-)$ denotes the Hochschild cochain complex. The product on $A_{\widehat{\infty}}$ comes from the product on $\operatorname{End}_{\mathrm{k}}(A)/A^{*}\otimes A$.

To describe the DG algebra $A_{\widehat{\infty}}$ more conceptually, we recall the following notion.

We can consider $A$ (and any other right $A$-module) as an object of $\operatorname{Rep}(A^{\mathrm{op}},\operatorname{Calk}_{\mathrm{k}})$ – suitably defined category of representations of $A^{\mathrm{op}}$ in $\operatorname{Calk}_{\mathrm{k}}$. Note that

The DG category of topological perfect complexes over $A_{\widehat{\infty}}$ is defined as follows.

Here the embedding $\operatorname{Rep}(A^{\mathrm{op}},\operatorname{Calk}_{\mathrm{k}})\hookrightarrow\operatorname{Perf}(A\otimes\operatorname{Calk}_{\mathrm{k}})$ comes from the assumption that $A$ is smooth. The functor $\operatorname{Perf}(A\otimes\operatorname{Calk}_{\mathrm{k}})\to\operatorname{Calk}_{A}$ is given by the tensor product: $(A,V)\mapsto V\otimes A$ for $V\in\operatorname{Calk}_{\mathrm{k}}$.

Alexander I. Efimov is a mathematician at the Steklov Mathematical Institute of RAS and also a member of the International Laboratory of Mirror Symmetry and Automorphic Forms, Higher School of Economics, Moscow, Russia. He works in algebraic geometry and is primarily interested in its categorical aspects, with relations to mirror symmetry, noncommutative geometry and $K$-theory. efimov@mccme.ru

References

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